Consider the parametric equations below.
x = sin2(t),    y = sin(2t),    0 ≤ t ≤ 
𝜋
2
Set up an integral that represents the area of the surface obtained by rotating the given curve about the x-axis.

To set up the integral representing the surface area obtained by rotating the parametric curve given by \( x = \sin^2(t) \), \( y = \sin(2t) \) about the \( x \)-axis, we'll use the surface area formula for a parametric curve rotated about the \( x \)-axis:

\[
A = \int_{a}^{b} 2\pi y \sqrt{\left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2} \, dt
\]

Given:
- \( x = \sin^2(t) \)
- \( y = \sin(2t) \)
- The parameter range is \( 0 \leq t \leq \frac{\pi}{2} \)

### Step 1: Compute \( \frac{dx}{dt} \) and \( \frac{dy}{dt} \)
- \( \frac{dx}{dt} = \frac{d}{dt} \left( \sin^2(t) \right) = 2\sin(t)\cos(t) = \sin(2t) \)
- \( \frac{dy}{dt} = \frac{d}{dt} \left( \sin(2t) \right) = 2\cos(2t) \)

### Step 2: Set up the surface area integral
Substitute the values of \( y \), \( \frac{dx}{dt} \), and \( \frac{dy}{dt} \) into the surface area formula:

\[
A = \int_{0}^{\frac{\pi}{2}} 2\pi \sin(2t) \sqrt{\left( \sin(2t) \right)^2 + \left( 2\cos(2t) \right)^2} \, dt
\]

### Step 3: Simplify the square root expression
The term inside the square root simplifies as follows:

\[
\left( \sin(2t) \right)^2 + \left( 2\cos(2t) \right)^2 = \sin^2(2t) + 4\cos^2(2t)
\]

Thus, the integral becomes:

\[
A = \int_{0}^{\frac{\pi}{2}} 2\pi \sin(2t) \sqrt{\sin^2(2t) + 4\cos^2(2t)} \, dt
\]

This is the integral representing the surface area of the curve rotated about the \( x \)-axis.

Would you like me to proceed with further simplification or evaluation of this integral?

### Related questions:
1. What is the physical significance of the surface area obtained by rotating a curve?
2. How does the parametric representation affect the surface area calculation for rotational surfaces?
3. Can surface area integrals be evaluated for curves rotated around the \( y \)-axis instead of the \( x \)-axis?
4. How would you handle the surface area if the limits of the parameter \( t \) were different?
5. How does the shape of the curve affect the complexity of the surface area integral?

### Tip:
When solving problems involving surface areas of rotation, always check if there's symmetry that can simplify the integral before proceeding with full evaluation.

Consider the parametric equations below.
x = sin^2(t),    y = sin(2t),    0 ≤ t ≤ π/2.
Set up an integral that represents the area of the surface obtained by rotating the given curve about the x-axis.

Learn how to set up the integral for finding the surface area of a curve defined by parametric equations x = sin^2(t) and y = sin(2t), rotated about the x-axis. This problem involves concepts of parametric equations, integral calculus, and surface area of revolution, suitable for advanced high school or early college students.

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